E0005E - Industrial Image Analysis

Transcription

1 E0005E - Industrial Image Analysis The Hough Transform Matthew Thurley slides by Johan Carlson 1

2 This Lecture The Hough transform Detection of lines Detection of other shapes (the generalized Hough transform) 2

3 Problem formulation Let s say we have an edge detected image. How do we put this into some practical use, for example automatically detecting shapes and object sizes? Original grayscale image Edge detected image Detected shape with dimensions?? 3

4 Problem formulation (cont d...) The first step is usually to find the edges. Original grayscale image, A Edge detected and thinned image, E This can be done for example by using a gradient edge detector, followed by some thresholding and morphological processing (such as thinning the lines by erosion or opening). 4

5 The Hough transform The result of applying the Hough transform will be a set of parameterized lines Edge detected and thinned image, E The result of Hough transform of E Line-crossings indicate corners Ended lines indicate objects 5

6 The Hough transform - line equation Recall that a straight line (in Cartesian coordinates) can be expressed as y = k x + m, where k is the gradient and m is the intersection with the y-axis. 6

7 The Hough transform - objects and lines Here, our original image has been transformed into a set of three parameterized lines. These can be used to characterize the detected object as a triangle. 7

8 The Hough transform - points on lines But how do we produce these parameterized lines? Let s start by considering an example with relatively few points, where three points lie on a line and the fourth doesn t. 8

9 The Hough transform - (m,k) space We know the line parameterization y = k x + m, where x and y are the variables, and k (gradient) and m (intersection) are constants. Let s think backwards, and instead express the equation as m = x k + y, and now let x and y be constants (i.e. a fixed point) and let m and k be the variables. We can think of this as being a transformation from (x, y)-space to (k, m)-space. 9

10 The Hough transform - (m,k) space Our four fixed points (1,3), (2,2), (4,0), and (4,3) can be plotted in (k, m)-space The intersection of the three lines in (k, m)-space is m = 4, k = 1, which is the parameters for the line in (x, y)-space. 10

11 The Hough transform - (m,k) space Problem The (k, m)-space cannot be used. The reason is that vertical lines can not be represented (since the gradient k ). Solution Let s use polar coordinates, meaning we represent each line as an angle θ and the perpendicular distance to the origin r. 11

12 The Hough transform - polar coordinates Our new parameter space will be (r, θ), where θ is the rotation angle and r is the perpendicular distance to the y-axis. 12

13 The Hough transform - polar coordinates A line in the (x, y)-space will thus be represented by a point in (r, θ)-space. 13

14 The Hough transform - polar coordinates 14

15 The Hough transform - An example 15

16 The Hough transform - polar coordinates Observation All points on a line will have the same values of the (r, θ) parameters. Problem However, for any single point, infinitely many lines will pass through it, if we allow the (r, θ)-space to be continuous. Solution Discretize the (r, θ) space, i.e. for each points in the image, only compute the parameters for a finite set of angles θ = θ 1, θ 2,...,θ N. For each θ i, we then obtain r i = xcos θ i + y sin(θ i ) 16

17 The Hough transform - accumulator matrix The accumulator matrix (array) Create a matrix, called the accumulator matrix, where each column corresponds to the angles θ = θ 1, θ 2,...,θ N and where the rows corresponds to bins or intervals of the resulting distances r. Then, for each point in the image, compute the r, θ values and increase the values of the corresponding elements of the accumulator matrix. Once we re done, the elements in the accumulator matrix corresponding to the highest values will correspond to lines in the image. 17

18 The Hough transform - accumulator matrix The accumulator matrix (cont d...) To see why, lets consider the following case All three points have one (r, θ) pair in common. This (r, θ) entry of the accumulator matrix will thus be 3. The other lines will result in the value 1 in the accumulator matrix. y x 18

19 The Hough transform (cont d...) Our original example Edge detected image Resulting accumulator matrix 19

20 The Hough transform (cont d...) Our original example (cont d...) 20

21 Detection of other shapes The generalized Hough transform The Hough transform can be generalized to find other shapes, e.g. circles or ellipses. However, the computational complexity increases drastically. Let s consider the example of finding a circle in an edge detected image. 21

22 Detection of other shapes (cont d...) The generalized Hough transform (cont d...) The equation of a circle can be written as (x x 0 ) 2 + (y y 0 ) 2 R 2 = 0, where (x 0, y 0 ) is the center of the circle (in Cartesian coordinates) and R is its radius. In this case, the Hough transform will be a transformation from the (x, y)-space to the (x 0, y 0, R)-space (i.e. to a three-dimensional space). 22

23 Detection of other shapes (cont d...) The generalized Hough transform (cont d...) Let s look at the simple case of five discrete points. Also, assume that we know the radius R = 1/ 2, thus reducing the dimension of the parameter space to two. 23

24 Detection of other shapes (cont d...) The generalized Hough transform (cont d...) For any fixed point (x, y) all possible circles with radius R = 1/ 2 passing through the point will contribute to the accumulator matrix. 24

25 Detection of other shapes (cont d...) The generalized Hough transform (cont d...) Somewhere (at some distinct point (x 0, y 0, 1/ 2)) in the accumulator matrix, the largest number of corresponding circles (in the parameter space) will coincide. This is the most probable location of the circle. 25

26 Summing Up Consider the following three questions; What do i need to work on? What have I learnt today? What was the main point left unanswered? Write your answers in the journal with the lecture number at the top of the page. 26